Near free-edge stresses in FRP-to-concrete bonded joint

Scientia Iranica A (2020) 27(6), 2727{2739

Sharif University of TechnologyScientia Iranica

Transactions A: Civil Engineeringhttp://scientiairanica.sharif.edu

Near free-edge stresses in FRP-to-concrete bondedjoint due to mechanical and thermal loads

H. Samadvand and M. Dehestani�

Faculty of Civil Engineering, Babol Noshirvani University of Technology, Babol, P.O. Box 484, Iran.

Received 14 March 2018; received in revised form 24 September 2018; accepted 24 December 2018

KEYWORDSFRP-to-concretebonded joint;Thermomechanicalload;Free-edge interfacialstress;Boundary layer;Debonding failure;Finite elementmethod.

Abstract. Over the last few decades, a considerable number of theoretical andexperimental investigations have been conducted on the mechanical strength of compositebonded joints. Nevertheless, many issues regarding the debonding behavior of such jointsstill remain uncertain. The high near free-edge stress �elds in most of these joints arethe cause of their debonding failure. In this study, the performance of an externallybonded Fiber-Reinforced Polymer (FRP) composite to a concrete substrate prism jointsubjected to mechanical and thermomechanical loadings was evaluated using the principlesof lamination theory. An inclusive Matlab code was generated to perform computations.The bond strength was estimated to take place in a region-also termed as the boundarylayer-where the peak interfacial shearing and transverse peeling stresses occurred, whereasthe preceding stress �eld was observed to be the main failure mode of the joint. Theproposed features were validated by using the existing experimental data points as well asthe commercial Finite Element (FE) modeling software, Abaqus. A comparison betweenthe calculated and experimental results demonstrated favorable accord, producing quitea high average ratio. The current approach facilitates failure modeling analysis, optimaldesign of bonded joints, and scaling analysis among others.

© 2020 Sharif University of Technology. All rights reserved.

1. Introduction

Bonded joints are popularly preferred over mechanicalbolts, rivets, and welds due to their compact design,low manufacturing cost, enhanced mechanical durabil-ity, sound/noise proof, and low weight. Accordingly,they are extensively used to transfer load on structuralelements and connections of civil structures [1]. Fur-thermore, the problem of function degradation of suchstructures has turned into a serious concern over the

*. Corresponding author. Fax: +981132331707E-mail addresses: [email protected] (H. Samadvand);[email protected] (M. Dehestani)

doi: 10.24200/sci.2018.50594.1779

past several years [2,3]. Meanwhile, thermal stressesplay a very important role in studying bonded jointsfor a variety of reasons. On the one hand, thesejoints receive considerable residual thermal stressesfrom the fabrication process. On the other hand,the materials that behave well at elevated or lowtemperatures should be selected. Thus, adherendsshould retain their properties at such temperatures;this is particularly true about the composite materials.The adherends with a speci�c Coe�cient of ThermalExpansion (CTE), possibly zero, are preferred [4].

From a historical point of view, Timoshenkodeveloped the concept of free-edge stress and �rstconsidered the de ection of bimetal thermostats sub-jected to uniform changes in temperature [5]. Then,continuous shear springs were applied to bonded joints

2728 H. Samadvand and M. Dehestani/Scientia Iranica, Transactions A: Civil Engineering 27 (2020) 2727{2739

in order to determine the maximum singular stressesat the edges [6]. This joint model is characterizedby simplicity, although the role of normal stresses inthe failure criterion is ignored. However, the afore-mentioned joint model and the associated interfacialstress solution have been adopted in many populartextbooks to perform the design and strength analysisof these types of joints. Following the introduction ofclassic models, there have been a signi�cant numberof theoretical complements to them. For instance, ane�cient adhesively bonded joint model based on theelementary Euler-Bernoulli beam theory was formu-lated [7] with emphasis on all mechanical propertiesand geometries of the adhesively bonded joints, inwhich the adherends were treated as Euler-Bernoullibeams and the adhesive layer was ignored due to itsnegligible deformation across the thickness. So far,with the fast development of microelectronics since the1970s, thermal stress-induced debonding failure of theso-called packaging has become one of the main tech-nical concerns and has attracted signi�cant academicinterest over the last few decades. For instance, aninnovative bonded joint model was developed withinthe framework of 2D elasticity in [8]. This modelpredicted the reasonable location of the peak interfacialshear stress at a distance of about 20% from the freeend of the adherend, without considering the normalstress �eld. An analytical solution of the stress-strainrelation of a bonded joint was developed in [9], fromwhich a parametric study of the overlap length andthickness of layers was carried out.

Since the 1990s, several improved methods forconducting stress analysis of bonded joints have beenreported in the literature. For instance, engineeringsolutions to the singular stress �eld near the interfaceedge of bimaterials were proposed in [10]. Furthermore,an analytical elastoplastic model of bonded joints waspresented, which was then veri�ed by using FiniteElement Method (FEM) [11]; Sayman also conductedan analytic study and concluded that the strengthof adhesively bonded joints could be increased byintroducing proper residual stresses. A model oflayer-wise adhesively bonded joints was formulated,from which the divided sub-layers were treated as�eld variables and solved through a set of governingOrdinary Di�erential Equations (ODEs) by evoking thetheorem of minimum strain energy [12]. An experi-mental investigation [13] into the suitability of CarbonFiber-Reinforced Polymer (CFRP) for strengtheningthe concrete-�lled steel tubular members under exurewas conducted. It was shown that the presenceof CFRP on the outer limits signi�cantly increasedthe performance of beams. The e�ciency of near-surface-mounted method for both exural and shearstrengthening of Reinforced Concrete (RC) beams wasexamined by applying an innovative manually-made

CFRP bar through experimental and numerical inves-tigations [14]. They de�ned 8-noded solid elements torepresent concrete beams and introduced link elementsfor modeling FRP by Ansys software. It was indicatedthat using the proposed bars signi�cantly improved theshear capacity of de�cient concrete beams. In a peerstudy, the exural performance of FRP-strengthenedRC beams under cyclic loads was examined to assessits exural behavior [15]. Eight RC beams undermonotonic and cyclic loadings were tested and theresults were presented in the form of load-de ectioncurves, reporting initiation and propagation of theinterfacial debonding at mid-span for specimens withend-anchor plates. In a partly similar experimentalstudy, the debonding mechanism for FRP-strengthenedRC beams within the framework of FE was studiedin [16]. After investigating the e�ects of lengthand width of the strengthening sheet on the beam'sbehavior and its failure mechanism, it was provedthat longer FRP sheets increased the load-carryingcapacity. Moreover, analysis of a membrane elementof an FRP-strengthened concrete beam-column jointwas proposed in [17]. The e�ects of bonding of steel,FRP, and concrete were considered in predicting theshear strength. It was found that even a low quantityof FRP could enhance the shear capacity of a joint byincreasing the con�nement. In addition, shear-torsioninteraction and cracking load of RC beams via FRPsheets were investigated in [18]. Lower-strengthenedspecimens were discovered to have undergone a greatercapacity increase. Furthermore, the exural andimpact behaviors of Glass Fiber-Reinforced Polymer(GFRP) laminates exposed to elevated temperatureswere investigated in [19]. It was pointed out thatthe exural and impact properties of laminates werereduced by increasing temperature. They also observedthat laminates with unidirectional �bers had the bestperformance subjected to elevated temperatures. Amicro-modeled computational framework for simulat-ing tensile response and tension-sti�ening behaviors ofFRP-strengthened RC elements was presented in [20].The bond stresses at the FRP-to-concrete interfacewere then computed based on the local stress transfermechanisms. An FE model to characterize the inter-facial shear strength between polymer matrices andsingle �laments was developed in [21]. A statisticalstudy was then followed to evaluate the in uenceof key geometrical test parameters on the variabilityof interfacial shear strength values. The RC beamspecimens strengthened by strips with bonded CFRPfan-shaped end anchorages to tension region wereexamined in [22]. It was indicated that the numberof anchorages applied to the ends of CFRP stripswas more e�ective than the width of CFRP strips,concerning strength and sti�ness of the specimens. Anumerical solution to calculating the interfacial stress

H. Samadvand and M. Dehestani/Scientia Iranica, Transactions A: Civil Engineering 27 (2020) 2727{2739 2729

distribution in beams strengthened with the FRP platewith the tapered ends subjected to thermal loading wasdeveloped by applying the Finite Di�erence Method(FDM) in [23]. They investigated the e�ects of di�erentpro�les on stress concentration reduction. Yet, theassumptions of parabolic through-thickness shear stresswere simpli�ed, and correction factors were consideredin the methodology.

Some researchers have conducted a thermal anal-ysis on the hybrid GFRP-concrete deck [24]. Theytheoretically investigated the CTE of GFRP laminateson a micro scale. Based on this analysis, the thermalbehavior of composite girders with a hybrid GFRP-concrete deck was studied on a macro scale, from whichthe results were more conservative than those of FEMoutputs. Others developed a combined experimental-numerical approach to characterizing the e�ect ofcyclic-temperature environment on bonded joints [3].Experimental tests were performed on single-lap jointswith CFRP and steel adherends in a cyclic-temperatureenvironment. It was then con�rmed that adhesivelybonded joints were allowed to have more uniform stressdistribution, joint complex shapes, and low structuralweight. An externally bonded CFRP plate used to rein-force the concrete beam was utilized in order to predictthe interfacial stresses in the adhesive layer account-ing for various e�ects of Poisson's ratio and Young'smodulus of the layer [25]. However, the adherendshear deformations were assumed to be due to theparabolic shear stress through thickness of the layers.Furthermore, a higher-order method was used in [26]to study thick composite tubes under a combination ofloads. By using the mentioned analytical approach,the displacement �eld of elasticity for a laminatedcomposite tube was obtained to calculate stresses. Inaddition, a higher-order theory was used for conductinghygrothermal analysis of laminated composite platesin [27], satisfying the inter-laminar shear stress conti-nuity at the interfaces and the zero transverse shearstress conditions at the top and bottom of the plate.They employed a 9-noded C0 continuous isoparametricelement to simulate the FE model.

In spite of the many research works done andwidespread use of regression analysis and �tting,some issues still remain unresolved when applying theabove adhesively bonded joint models for determin-ing the stress �eld in composite bonded joints, inwhich stress singularities occur at the end of bindinglines and laminate interfaces. As a matter of fact,transverse deformation plays a critical role in de-lamination failure of composite joints, in which thetransverse modulus of composite laminates is verylow.

1.1. Research attributesThe main objective of this work is to establish two in-

terfacial shear and transversely normal stress functionsby integrating the geometric, mechanical, and thermalparameters of the joint under consideration in order toaccurately determine the stress �eld distributions alongthe interface of the CFRP cover and concrete substratelayers. The salient feature of the approach is that itsatis�es all the Boundary Conditions (BCs) in the jointexactly as necessitated by the theoretical framework,a feature ignored by many related preceding works,even the renowned ones. Furthermore, the approachexpresses two simple and explicit equations for de�ningthe interfacial shear and transversely normal stress�elds at any point of the bonding line. In addition,this approach can be employed to extend the scopeof failure modeling analysis, optimal design of bondedjoints, and scaling analysis by altering the type ofloading, joint con�gurations, material type, etc. More-over, the application of this approach casts light onthe problem without conventional oversimpli�cations,leading to the closed-form determination of the citedstress �elds.

2. Stress analysis through thermo-elasticlamination theory

In this section, the determination of the interfacialshear and normal stresses of the externally-bondedFRP-to-concrete joints subjected to mechanical andthermomechanical loads is discussed in detail. Thestress analysis conducted is based on the mechanics ofthe composite materials approach incorporated by thee�cient algorithms in Matlab software. The generatedcomputational code embraces all the mechanical, ther-mal, and geometric properties of layers, in addition tothe external loadings of the joint.

In order to develop the stress-strain relationshipfor the layers of the joint, the 2D constitutive equationfor each layer must be established as Eq. (1):24�x�y

�xy

35 =

24 �Q11 �Q12 �Q16�Q21 �Q22 �Q26�Q61 �Q62 �Q66

3524 "x � "Tx"y � "Ty xy � Txy

35 ; (1)

where �Qij is the transformed reduced sti�ness coe�-cient. The [ �Q] can be expressed by Eq. (2):� �Q

�= [T ]�1[Q][R][T ][R]�1; (2)

where [T ] is termed the transformation matrix and isde�ned by Eq. (3):

[T ] =

24 cos2 � sin2 � 2 sin � cos �sin2 � cos2 � �2 sin � cos �

� sin � cos � sin � cos � cos2 � � sin2 �

35 :(3)

[R] is the Reuter matrix. The inverse of compliance


matrix is [Q], the sti�ness matrix, i.e., [Q] = [S]�1.The compliance matrix, [S], can be expressed in Eq. (4)in terms of the engineering constants of the material,which can be written as follows:

[S] =

24S11 S12 S16S21 S22 S26S61 S62 S66

35=

24 1=E1 ��12=E1 0��12=E1 1=E2 0

0 0 1=G12

35 : (4)

To determine the strains in Eq. (1), Eq. (5) is read asfollows:

8<: "x"y xy

9=; =

8>>>><>>>>:"0x

"0y

0xy

9>>>>=>>>>;+ z

8<: �x�y�xy

9=;+

8>>>><>>>>:"Tx

"Ty

Txy

9>>>>=>>>>; ; (5)

where "0, �, and "T terms are the midplane strains,curvatures, and thermal strains, respectively; Eq. (5)is considered to be the fundamental equation of lami-nation theory. The total strains comprise superpositionof the midplane strains, the strains associated withcurvature, and the strains originating from thermale�ects. It is worth noticing that this result wasachieved without determining the type of materialin advance or knowing whether or not the layer iscomposed of more than one ply. The mentioned resultwas derived from Kirchho�'s assumptions about thedisplacements, considering that the direction normalto the midplane remained straight and normal to thedeformed midplane after deformation; the directionnormal to the midplane did not change length; and thedisplacements were independent of material considera-tions [4].

Now, only if the midplane and thermal strainsand curvatures are known, the stress state can bedetermined using the constitutive Eq. (1). The z inEq. (5) is the point of interest at which the stresses andstrains are calculated and measured from the midplane.The midplane strains and curvatures can be measuredusing Eq. (6) as follows:8>>>>>><>>>>>>:

"0x"0y

0xy�x�y�xy

9>>>>>>=>>>>>>;=�

[A] [B][B] [D]

�8>>>>>><>>>>>>:

Nx +NTx

Ny +NTy

Nxy +NTxy

Mx +MTx

My +MTy

Mxy +MTxy

9>>>>>>=>>>>>>;; (6)

where [A], [B], and [D] matrices denote extensional,coupling, and bending compliance matrices, respec-tively, and are de�ned by Eq. (7):

8>>>>>>>>>><>>>>>>>>>>:

Aij =nPk=1

� �Qij�k tk i = 1; 2; 6

Bij = 1=2nPk=1

� �Qij�k t

2k i = 1; 2; 6

Dij = 1=3nPk=1

� �Qij�k t

3k i = 1; 2; 6

(7)

where tk is the thickness of the kth layer. Theresulting forces and moments are expressed in thefollowing. Nx and Ny are normal forces, and Nxy isthe shear force per unit length. Likewise, Mx andMy are the bending moments, and Mxy is the twistingmoment per unit length. Hence, the only remainingconsideration for thermo-elastic lamination theory isthe utilization of the equivalent thermal force, NT , andthe equivalent thermal moment, MT . These quantitiesare titled equivalent forces and equivalent moments,respectively, since they have units of force and momentper unit length and are not physically applied forcesand moments.

3. Problem modeling

Figure 1(a) and (b) show the con�guration of thecomposite bonded joint comprising a straight CFRPcover layer and a concrete substrate bar, together

Figure 1. Con�guration of the joint consisting of astraight CFRP cover and a concrete substrate withdistribution of near free-edge interfacial stresses: (a)Bonded joint and (b) distribution of interfacial shear (�)and normal (�) stresses.


with the resulting interfacial shear (�) and normal(�) stresses, where the bonding line is denoted by L.The beam is assumed to be subjected to the action ofuniform axial tension, P0, which is applied to both endsof the tension bar, namely the concrete substrate layer.Subscripts 1 and 2 are designated to the cover layerand substrate bar, respectively. The layers enjoy themechanical properties of Young's modulus Ei (i = 1; 2)and Poisson's ratio �i (i = 1; 2) as well as the geometricparameters of thickness hi (i = 1; 2) and CTE �i(i = 1; 2). The width is denoted by b. The x-coordinatein a 2D model is taken from the mid-span of the joint;the z-coordinate is selected perpendicular to the x-coordinate and along the layer edge.

The adherends of bonded joints are in a compli-cated three-dimensional stress state. This is mainly dueto the varying Poisson's ratios across the adherends'interface. To facilitate the analysis process, the jointwas considered to be in the plane-stress state andsubjected to a uniform temperature change �T relativeto the reference temperature. The interfacial shearand normal stresses might result from the mismatchof material properties along the bonding line; lateralde ection was probably induced since the joint wasnot laterally symmetric. The joint layers were treatedas isotropic for the concrete substrate bar and trans-versely isotropic for the CFRP cover layer and linearlythermoelastic materials.

Thereafter, based on the commercial Finite Ele-ment Analysis (FEA) software package Abaqus, FEMwas employed to further authenticate the proposedcomposite approach. Through the course of apply-ing FEM for the joint, the 8-noded linear elements(C3D8R) and exclusively hexahedral meshes were em-ployed, as well. A nonuniform biased distribution ofelements along the bondline was used to exhibit thetrend of singular stresses along the bondline, as shownin Figure 2. In addition, as illustrated in Figure 3, twoof the substrate ending edges were used to constrainthe movement of the selected degrees of freedom towardzero so as to simulate the BCs.

The whole solving process was executed by pro-graming a robust Matlab code encompassing all the

Figure 2. Representative of the biased mesh grid densityof bonded joint.

Figure 3. Loading and Boundary Conditions (BCs)applied to ending edges of the concrete substrate.

mechanical properties, geometries, and external ther-mal load of the joint. Consequently, the entire stress�eld of the bonded joint can be determined accurately,which is validated by the existing models in theliterature. The detailed work ow of the stress coe�-cients derivation is given in Figure 4. It characterizesthe incremental analysis procedure owchart for theFRP-to-concrete composite bonded joint model underconsideration.

4. Governing equation of interfacial stressfunctions

Figure 5(a) and (b) illustrate the typical representativesegmental elements of the reinforcing CFRP-laminateand the concrete substrate layer together with thestress resultants, conforming to the conventional signstandards [28]. Therefore, as in the representative seg-mental element of the reinforcing patch (Figure 5(a)),two of static equations of equilibrium are expressed byEqs. (8) and (9):

dQ1

dx= ��b; (8)

dM1

dx= Q1 � �h1b

2: (9)

Meanwhile, the equivalent equilibrium equations forthe representative segmental element of the concretesubstrate layer are formulated as Eqs. (10) and (11):

dQ2

dx= �b; (10)

dM2

dx= Q2 � �h2b

2: (11)

Two unknown stress functions including the interfacialshear and normal stresses, as denoted by �(x) and �(x),were adopted. Since the joint was symmetric withrespect to y-axis, the shear stress function �(x) andthe normal stress function �(x) would be odd and evenfunctions, respectively.

Furthermore, the x-axis was aligned horizontallyso that the two unknowns could be expressed as thefunctions of variable x. The shear stress at the freeedges of the bonding line, x = �L=2 and x = +L=2,reduced the interfacial shear stress function to zero.


Figure 4. Incremental procedure to determine interfacial stress coe�cients of the bonded joint.

Integration of Eqs. (8) and (9) with respect to xfrom x = �L=2 as well as application of the associatedBCs, i.e., Q1(�L=2) = 0 and M1(�L=2) = 0, led toEqs. (12) and (13), respectively:

Q1(x) = �bxZ

�L=2�(�)d�; (12)

M1(x)=�bxZ

�L=2

�Z�L=2

�(�)d�d�� h1b2

xZ�L=2

(�)d�: (13)

The same trend was used for the resulting stresses ofthe concrete substrate by integrating Eqs. (10) and (11)

with respect to x from x = �L=2 and taking intoaccount the relevant BCs, i.e., Q2(�L=2) = 0 andM2(�L=2) = 0, for the shear and moment, respectively.The shear force, Q2(x), and bending moment, M2(x),can be simply determined using Eqs. (14) and (15) asfollows:

Q2(x) = bxZ

�L=2�(�)d�; (14)

M2(x) = bxZ

�L=2

�Z�L=2

�(�)d�d� � h2b2

xZ�L=2

(�)d�:(15)

In order to correlate the interfacial shear �(x) and


Figure 5. Free-body diagrams of representative segmental elements of the adherends: (a) Carbon Fiber ReinforcedPolymer (CFRP)-composite cover and (b) concrete substrate layer.

normal �(x) stress functions along the interface andto simplify the calculations, the deformation compati-bility, as represented by Eq. (16), was used such thatthe radii of curvature of CFRP-composite laminateand the concrete substrate layer were assumed to beapproximately the same:

1�1

=1�2� M1

E1I1=

M2

E2I2; (16)

where I1 = 1=12bh31 and I2 = 1=12bh3

2 are the momentsof inertia of the CFRP-laminate composite and theconcrete substrate layer, respectively.

Hence, substituting Eqs. (13) and (15) intoEq. (16) becomes:

�E2

E1

�h2

h1

�3264 xZ�L2

�Z�L2

�(�)d�d� � h1

2

xZ�L2

�(�)d�

375=

xZ�L2

�Z�L2

�(�)d�d� � h2

2

xZ�L2

�(�)d�: (17)

By di�erentiating both sides of Eq. (17) and de�ningthe parameters as e21 = E2=E1, h21 = h2=h1, and�0 = e21h3

21 + 1=(e21h321 � 1), Eq. (17) can be reduced

to Eq. (18) as follows:

�(x) = �2�0

h2

xZ�L=2

�(�)d�: (18)

Then, the entire strain energy of the CFRP-to-concretejoint is determined by integrating the strain energydensity (per unit length) with respect to the x fromx = �L=2 to x = +L=2, as given in Eq. (19):

U =b

L2Z

�L2

h1=2Z�h1=2

�12

h�(1)xx "

(1)xx + �(1)

yy "(1)yy

i+

1 + �1

E1

�� (1)y1x

�2�dxdy1

+ b

L2Z

�L2

h2=2Z�h2=2

�12

h�(2)xx "

(2)xx + �(2)

yy "(2)yy

i+

1 + �2

E2

�� (2)y2x

�2�dxdy2: (19)

The aforementioned complementary strain energy isa functional with respect to the unknown interfacialstress function �(x).

Based on the theorem of minimum complemen-tary strain energy, the complementary strain energy ofthe joint with respect to the interfacial stress function,�(x), reaches a stationary point in static equilibriumsuch that @U = 0, where � is the mathematical varia-tional operator with respect to the unknown interfacialshear stress, �(x). By setting it equal to zero, the 4th-order ODE was yielded with respect to the interfacialshear stress �(x) as Eq. (20):

IV (�)� 2pII(�) + q2(�) + t = 0: (20)

In Eq. (9):

(�) = �xh1

�= �

�1

P0h1

� xZ�L=2

�(x)dx (21)

is a dimensionless stress function; the coe�cients p =B=(�2A), q =

pC=A, and t = D=A are related to

the properties of the joint. Moreover, each of the


coe�cients A, B, C, and D is in turn solved by usingan e�cient code generated by the authors in Matlab,as elaborated in Figure 4.

Mathematically, in the case of q > p, the solutionto Eq. (9) gives the following:

(�) =C1 cosh(��) cos(!�) + C2 sinh(��) sin(!�)

+ t=q2; (22)

where ! =p

(q � p)=2, � =p

(p+ q)=2, and C1 andC2 are constants. By applying the shear-free and axialtraction-free conditions at x = �L=2, C1 and C2 canbe determined and �(x) can be expressed as Eq. (23):

�(x) =� P0h1d(�)dx

=� P0 [(C1�+ C2!) sinh(�x=h1) cos(!x=h1)

+(�C1!+C2�) cosh(�x=h1) sin(!x=h1)] :(23)

In addition, if p > q, the solution to Eq. (9) yields thefollowing:

(�) = C1 cosh(��) + C2 cosh(!�) + t=q2; (24)

where ! =qp�pp2 � q2, � =

qp+

pp2 � q2, and

C1 and C2 are two constants. By plugging the shear-free and axial traction-free conditions at x = �L=2, C1and C2 can be determined and �(x) can be expressedas in Eq. (25):

�(x) =� P0h1d(�)dx

=�P0 [C1� sinh(�x=h1)+C2! sinh(!x=h1)] :(25)

Accordingly, the interfacial normal stress �(x) relatedto �(x) through Eq. (18) can be de�ned as in Eq. (26):

�(x) = � h2

2�0� 0(x): (26)

5. Validating process

In order to validate the present model, herein, thethermomechanical stress analysis of a bimaterial ther-mostat subjected to a uniform temperature change [29{31] was considered. Consistent with the coordinatesystem depicted in Figure 1, the parameters:

h1 = 2:5 mm; E1 = 70 GPa; �1 = 0:345;

�1 =23:6e� 61=C; h2 =2:5 mm; E2 =325 GPa;

�2 = 0:293; �2 = 4:9e� 61=C; L = 50:8 mm;

and �T = 240�C;

where �i is the CTE with respect to the principalmaterial directions.

By substituting the aforementioned values intoEqs. (25) and (26) and determining interfacial shearand normal stresses of the joint subjected to mechanicaland thermomechanical loads for a joint model with agiven set of parameters, the analysis includes each ofthe resulting interfacial shear and normal stress �eldsthat can be expressed explicitly through the simple andstraightforward Eqs. (27) and (28), respectively:

�xz =4:2328� 10�015 sinh(1:5631x)

� 2:2095� 10�009 sinh(1:0448x); (27)

where �xz is the interfacial thermal shear stress and:

�zz =1:8629� 10�009 cosh(1:0448x)

� 5:3392� 10�015 cosh(1:5631x); (28)

where �zz is the interfacial thermal normal stress alongthe bonding line.

Figure 6(a) and (b) show the distribution of

Figure 6. Comparison of the stresses proposed by thepresent model with those by Wu and Jenson and FEMdue to a uniform change of temperature �T : (a)Interfacial thermal shear stress (�) and (b) interfacialthermal normal stress (�).


the interfacial thermal shear and normal stress �eldsof the tested joint subjected to uniform changes intemperature �T , which were predicted by the presentlamination theory approach and compared with thoseobtained by FEM (Abaqus) and those in [32] along theinterface. It can be observed that the shearing andpeeling stresses are highly localized at the near free-end of the bondline. Obviously, the interfacial shearstresses predicted by the present lamination theoryapproach exactly satisfy the shear-free BCs in the joint,as required by the theoretical formulation.

Even though the normal and shear stresses ex-hibited very similar variation trends for the proposedapproach, FEM, and those of [32], shear stress resultsobtained by FEM did not satisfy the stress-free BCsat the very ending edges of the cover layer becauseof the singular state of such stresses. However, thepeak values took place at the same point as those ofthe lamination theory approach. The interfacial shearstress values were 25 and 35% lower at the adherendend than those predicted by Wu and Jenson [32] andFEM, respectively. This can be due to the lower-ordernature of the approach in the current work. Moreover,the peak values of normal stresses predicted by theformulations in the present study were 30% higherand 20% lower than those of [32] and FEM with thesame geometries, material properties, and temperaturechange. After employing the temperature change to thejoint, the cover layer showed greater thermal expansionand tended to endure larger deformations due to thelarge CTE than the substrate layer at a gross ratio of5 to 1.

6. Scaling analysis

In terms of scaling analysis, the present solution to theinterfacial stresses in bonded joints has more advan-tages than the conventional approaches such as simplic-ity and explicit expressions of the stress components. A

compact and e�cient computational Matlab code wasdesigned and implemented to examine the dependencyof the interfacial shear and normal stresses on thetemperature parameter, as well as other mechanicalvariables. Table 1 depicts the mechanical and geo-metric properties of concrete substrate in the scalinganalysis. Likewise, Table 2 displays the mechanicaland geometric properties of the CFRP composite coverlayer in this section.

To carry out the scaling analysis of thermome-chanical stresses in the joint, four positive and fournegative temperatures, ranging from �T = 25�C to�T = 100�C and �T = �25�C to �T = �100�Cat a span of 25 were adopted. Figure 7(a) and (b)and Figure 8(a) and (b) illustrate the variations in theinterfacial shear and normal stresses with distance xfrom the left end of the CFRP cover layer for positiveand negative thermal loads, respectively.

It is perceived that shear and normal stresses aresymmetric with respect to the mid-span of the joint.According to Figure 7(a) and (b), as the positive valuesof temperature increased, the interfacial shear stress�eld increased in the near free-edge zone. Speci�cally,the peak values of interfacial shear stress appearedat a distance close to free ends, also termed theboundary layer region. In this research, the distancewas measured to be the L=8 from the free edges. Theshear stress was equal to zero at free edges; on thecontrary, the peak values of the positive normal stressoccurred exactly at free ends. As shown in Figure 8(a)and (b), the increasing trend of stresses is the same, i.e.,absolute increase in temperature causes an increase inshear and normal stresses. It was then observed thatthe normal stresses took negative values for negativethermal loads. Due to the main contribution of shearstresses, the peak values of interfacial shear stresseswere much larger than those of the normal stressesin all the cases under examination, demonstrating thedominant mode of debonding failure in the joint.

Table 1. Mechanical and geometric properties of concrete substrate prism.

AxialmodulusE2

(GPa)

ShearmodulusG12

(GPa)

Poisson'sratio�2

f 0c concretecylinderstrength(MPa)

Axial CTE�1

(�/�C)

Thicknessh2 (mm)

Jointwidthb (mm)

BondlineL

(mm)

TractionloadP0

(MPa)

33.8 14.08 0.2 50 10 50 50 100 +1

Table 2. Mechanical and geometric properties of the Carbon Fiber Reinforced Polymer (CFRP)-composite cover layer.

CFRP

AxialmodulusE1

(GPa)

ShearmodulusG12

(GPa)

Poisson'sratio�1

AxialCTE

�1 (�/�C)

Thicknessh1

(mm)

Jointwidthb

(mm)

BondlineL

(mm)

TractionloadP0

(MPa)

T300/5208 132 5.65 0.24 {0.77 5 50 100 +1


Table 3. Mechanical and geometric properties of the joint layers.

CFRPaxial

modulusE1

(GPa)

Concreteaxial

modulusE2

(GPa)

CFRPthickness

h1

(mm)

Concretethickness

h2

(mm)

CFRPPoisson's

ratio�1

ConcretePoisson's

ratio�2

Jointwidthb

(mm)

BondinglengthL

(mm)

Thermalload�T(�C)

132 33.8 5 50 0.24 0.2 50 100 100

Figure 7. Variations in the interfacial shear and normalstresses with respect to positive thermal loads: (a)Interfacial shear stress (�) and (b) interfacial normalstress (�).

In order to evaluate the mechanical loading e�ect,four positive values of P0 = 1, 10, 20, and 30 MPawere adopted in the study. Table 3 illustrates thejoint material properties through the analysis. Thedistribution of the shear and normal stress �elds alongthe interface with respect to di�erent values of P0 isgiven in Figure 9(a) and (b). It was found that thestress �elds were symmetric with respect to the mid-span of the joint. The stress distribution was identicalto that of the thermal loading e�ect, though much

Figure 8. Variations in the interfacial shear and normalstresses with respect to negative thermal loads: (a)Interfacial shear stress (�) and (b) interfacial normalstress (�).

more intense. In both Figure 9(a) and (b), the stressesincrease as P0 escalates. Additionally, four negativevalues of P0 = �1, {10, {20, and {30 MPa wereassumed. Material properties are shown in Table 3, andFigure 10(a) and (b) plot variations of the interfacialshear and normal stresses along the bondline withrespect to negative traction loads, respectively. Asthe absolute value of P0 increased from P0 = �1 to{30 MPa, the interfacial stresses tended to decreasedown to a certain load, i.e., P0 = �19 MPa, whereas


Figure 9. Variations in the interfacial shear and normalstresses with respect to positive mechanical loads: (a)Interfacial shear stress (�) and (b) interfacial normalstress (�).

the curvature at this load level changed so that the peakinterfacial normal stresses would appear to be negativeafter this loading point, having an increasing trend.

Finally, Figure 11(a) and (b) depict the distri-bution of interfacial shear and normal stress �elds atvarying temperatures with respect to di�erent values ofP0, respectively. As the thermal load, �T , increasedfrom �T = �100�C to �T = 100�C, the shear stressdecreased. Thus, for a particular thermal load, shearstress decreased, while the mechanical traction loadincreased. This tendency goes against the normalstresses in a sense that the increase of traction loadfor a speci�c thermal load leads to an increase in thenormal stress �eld. According to all �gures, the stressvalues are zero at the points without any thermal ormechanical loads.

As acknowledged using the validating process(Section 4), the obtained results of the present studyshowed the same quantitative features as before andwere in sensibly acceptable agreement with the previ-ous investigations. Yet, the formalism can be readilyextended to the assortment of bonded structures by

Figure 10. Variations in the interfacial shear and normalstresses with respect to negative mechanical loads: (a)Interfacial shear stress (�) and (b) interfacial normalstress (�).

applying the robust computational code based on thepresented theoretical approach and introducing anadditional ply as the adhesive layer and plugging themechanical and geometric properties into the algorithmto consider the e�ects of the adhesive layer on thestress distribution. As a result, it has the capabilityto provide a framework for future studies.

7. Concluding remarks

The performance of an externally bonded FRP-to-concrete joint subjected to mechanical and thermalloadings was evaluated. A theoretically self-consistentlower-order lamination-theory approach was proposedto establish two interfacial shear and transverselynormal stress functions in terms of two simple andexplicit equations by integrating the geometric, me-chanical, and thermal parameters of the joint. Anenhanced understanding of the scaling behavior ofstress variations was obtained by devising a compactand e�cient computational Matlab code to examine


Figure 11. Stress �eld distribution along varyingtemperatures with respect to di�erent values of P0: (a)Interfacial shear stress (�) and (b) interfacial normalstress (�).

the dependency of the interfacial shear and normalstresses upon temperature, as well as other mechanicalvariables. The following conclusions were thus drawn:

1. Normal and shear stresses were characterized bythe very similar varying trends for the proposedapproach as to Finite Element Method (FEM), inthat the peak values took place at the same points,though shear stresses obtained via FEM did notsatisfy the stress-free Boundary Conditions (BCs)at the very ending edges of the bondline due tosingular nature of such stresses;

2. Compared to FEM, the proposed approach provedto be more competent for stress analysis of bondedjoints since it only calls for the input of basicdimensions and material properties;

3. Increase in temperature escalated both shear andnormal stresses;

4. The stress distribution subjected to mechanicalloading was identical to that of the thermal loadinge�ect, yet much more intense;

5. Shear stress was equal to zero at free edges; on thecontrary, the absolute peak value of normal stressoccurred precisely at free ends;

6. Peak values of interfacial shear stress appeared at adistance close to free ends, termed as the boundarylayer region. In this research, the length of thisregion was measured to be L=8 from the free edges;

7. Peak values of interfacial shear stresses were muchlarger than normal stresses in all the cases underexamination, demonstrating the dominant mode ofdebonding failure in the joint.

No funding source was provided in the conductionor preparation of the current research.

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Biographies

Hojjat Samadvand received his MSc in StructuralEngineering at Babol Noshirvani University of Tech-nology and is currently a PhD Candidate in Struc-tural/Earthquake Engineering at Urmia University.His major areas of interest are the mechanics ofreinforced concrete structures and nano-composite ma-terials, as well as numerical modeling. He has alreadypresented several articles in the leading internationaland national conferences on Civil Engineering includingICCE and NCCE.

Mehdi Dehestani is currently an Associate Professorof Civil Engineering at Babol Noshirvani University ofTechnology, Babol, Iran. He is also a faculty memberof the Structural and Earthquake Engineering at BabolNoshirvani University of Technology. He received hisPhD degree in Structural and Earthquake Engineeringfrom Sharif University of Technology, Tehran, Iran.His research expertise lies in the area of reinforcedconcrete structures, speci�cally the application of �ber-reinforced polymer composites and the fracture me-chanics of a variety of reinforced concrete components.He has already issued over 80 research papers with themost prominent journals and conferences in the �eld ofCivil Engineering.

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Near free-edge stresses in FRP-to-concrete bonded joint